The generator matrix

 1  0  0  0  1  1  1 X^2  1  1 X^3+X  X  1 X^3+X  1 X^3+X^2+X X^3  1 X^3+X^2  1  1  1 X^3+X  1 X^3  0  1  1  1 X^3+X  X  1 X^2  1 X^2  1  1  1
 0  1  0  0 X^2 X^3+1  1  1 X^2+1 X^3+1 X^3+X^2  1 X^3+X^2  1 X^3+X^2+X  1  1 X^3  X X^2+1 X^3+X X+1  1 X+1  1 X^3+X^2+X X^3 X^3+X^2+X X^3+1  1 X^3 X^2+X  1 X^3+X^2  1 X^2+1  X  0
 0  0  1  0 X^2+1  1 X^2 X^2+1 X+1 X^2+X  1 X^2 X^2+X+1 X+1  0 X^3+X X^3+X X^2+1  1 X^3+X X^3+X+1 X^3+X^2+1 X^3+X^2+X+1 X^2 X^3+X+1  1  X X^3+X^2+X X^3+X^2+1 X^2+1 X^2 X^3+1  1 X^3+X^2  X X^3+X^2+X X^3+X X^3
 0  0  0  1  1 X^2 X^2+1 X^3+1 X+1 X^2+X X^3+1 X^2+X+1 X^3  0 X^3+1 X^3+X^2 X+1  X X^3+X^2+X X^3+X^2+1  1 X^3+X+1 X^3+X^2+1 X^2+X X^3+X X^3+X^2+1 X^3+X^2 X^2+X+1 X^3+X^2+X  X  1 X^3+X+1 X^2+X+1 X^3+X+1 X^3 X^3+X+1 X^3+X X^3

generates a code of length 38 over Z2[X]/(X^4) who´s minimum homogenous weight is 32.

Homogenous weight enumerator: w(x)=1x^0+104x^32+1008x^33+2272x^34+5304x^35+7256x^36+10862x^37+11655x^38+11650x^39+7003x^40+5028x^41+2186x^42+852x^43+226x^44+94x^45+13x^46+18x^47+2x^48+2x^54

The gray image is a linear code over GF(2) with n=304, k=16 and d=128.
This code was found by Heurico 1.16 in 18.2 seconds.